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Designing Angle-Independent Structural Colors Using Monte Carlo

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Designing Angle-Independent Structural Colors Using Monte Carlo Designing angle-independent structural colors using Monte Carlo simulations of multiple scattering Victoria Hwanga , Anna B. Stephensona, Solomon Barkleyb, Soeren Brandta , Ming Xiaoa, Joanna Aizenberga,c, and Vinothan N. Manoharana,b,1 aHarvard John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138; bDepartment of Physics, Harvard University, Cambridge, MA 02138; and cDepartment of Chemistry and Chemical Biology, Harvard University, Cambridge, MA 02138 Edited by Eli Yablonovitch, University of California, Berkeley, CA, and approved December 6, 2020 (received for review July 23, 2020) Disordered nanostructures with correlations on the scale of visible Because structural colors do not require strong absorption, they wavelengths can show angle-independent structural colors. These resist photobleaching and are less photoreactive. Also, because materials could replace dyes in some applications because the structural colors arise from scattering, different colors can be color is tunable and resists photobleaching. However, designing made from the same component materials by changing the nanostructures with a prescribed color is difficult, especially when sizes of the pores or particles. This property enables a power- the application—cosmetics or displays, for example—requires spe- ful approach to formulation: First, the component materials for cific component materials. A general approach to solving this a particular application are chosen to meet constraints such as constrained design problem is modeling and optimization: Using low toxicity or reactivity; second, the nanostructure is tuned to a model that predicts the color of a given system, one opti- achieve the target color. mizes the model parameters under constraints to achieve a target Making this approach a reality requires a way to design color. For this approach to work, the model must make accurate materials with a target color, starting from a prescribed set predictions, which is challenging because disordered nanostruc- of component materials and, frequently, a particular type of tures have multiple scattering. To address this challenge, we nanostructure. Many studies on angle-independent color deal develop a Monte Carlo model that simulates multiple scatter- with a less-constrained design problem: Given a color, deter- ing of light in disordered arrangements of spherical particles or mine the materials or nanostructure required to make it. A voids. The model produces quantitative agreement with measure- common approach to address this less-constrained problem is ments when we account for roughness on the surface of the film, biomimicry. Examples of biomimicry include fabricating struc- ENGINEERING particle polydispersity, and wavelength-dependent absorption in turally colored materials containing melanin, an absorber found the components. Unlike discrete numerical simulations, our model in bird feathers (3), to increase the saturation (4–8) or mak- is parameterized in terms of experimental variables, simplifying ing nanostructures that mimic those found in butterfly wings the connection between simulation and fabrication. To demon- (9). But biomimicry is not a general approach to making struc- strate this approach, we reproduce the color of the male mountain turally colored materials. In some applications, a biomimetic bluebird (Sialia currucoides) in an experimental system, using pre- system may not be compatible with the constraints: For exam- scribed components and a microstructure that is easy to fabricate. ple, reflective displays might require nonabsorbing materials, Finally, we use the model to find the limits of angle-independent so that the display does not heat when illuminated. In other structural colors for a given system. These results enable an engineering design approach to structural color for many Significance different applications. Structural color comes from interference of light scattered structural color j Monte Carlo j multiple scattering j engineering design from a nanostructure. Disordered nanostructures have struc- tural colors that are independent of viewing angle, similar to ngle-independent structural color occurs when light scat- dyed materials. Unlike dyes, structural colors resist fading and Aters from a composite material with a correlation length can be broadly tuned, making them useful for many applica- on the scale of visible wavelengths. Examples of such materi- tions. However, making a nanostructure with a given color als are the feathers of blue and some green birds (Fig. 1A), as is challenging because there are so many tunable parame- well as disordered packings of colloidal particles (1) with radii ters. Furthermore, applications such as cosmetics or displays around 100 to 150 nm (Fig. 1B). The short-range correlations require specific component materials. To solve this design between the pores in the bird feathers (Fig. 1C) and the parti- problem, we develop a model that quantitatively predicts the cles in the colloidal sample (Fig. 1D) give rise to constructive color for given experimental parameters. We then use opti- interference of backscattered light over a broad range of scat- mization to determine the parameters required to make a tering wavevectors q (2). This broad range is directly responsible target color under specific constraints. This approach makes for the weak angle dependence of the colors, since the scatter- it possible to engineer structural color for many different ing wavevector jqj = 4π sin (θ=2)/λ couples the scattering angle applications. θ and the wavelength λ. In comparison to the sharp Bragg peaks Author contributions: V.H., A.B.S., J.A., and V.N.M. designed research; V.H., A.B.S., and that occur when light scatters from a material with long-range S. Brandt performed research; V.H., A.B.S., S. Barkley, M.X., and V.N.M. contributed new order, such as a colloidal crystal, the reflection of a material reagents/analytic tools; V.H. and A.B.S. analyzed data; and V.H. and V.N.M. wrote the with short-range order peaks at a lower intensity. Nonetheless, paper.y short-range order can give rise to vivid colors (Fig. 1A). Fur- Competing interest statement: A provisional patent application has been filed on the thermore, the weak angle dependence means that the color is subject of this work, with V.H., A.B.S., M.X., and V.N.M. as inventors.y almost indistinguishable from that produced by an absorbing dye This article is a PNAS Direct Submission.y or pigment. Published under the PNAS license.y We can, therefore, envision replacing traditional dyes and pig- 1 To whom correspondence may be addressed. Email: [email protected] ments with angle-independent structural colors in applications This article contains supporting information online at https://www.pnas.org/lookup/suppl/ ranging from paints, coatings, and cosmetics to electronic dis- doi:10.1073/pnas.2015551118/-/DCSupplemental.y plays and sensors. There are several advantages to doing so. Published January 20, 2021. PNAS 2021 Vol. 118 No. 4 e2015551118 https://doi.org/10.1073/pnas.2015551118 j 1 of 10 Downloaded by guest on September 26, 2021 A B G C D H E I F Fig. 1. Overview of design approach. (A and B) Photograph (A) and scanning electron micrograph (SEM) (B) of features from a male Abyssinian roller (Specimen MCZ:Orn:63369. Coracias abyssinica. Africa: Sudan: Blue Nile. El Garef. John C. Phillips; image credit: Museum of Comparative Zoology, Harvard University, C President and Fellows of Harvard College). (C) Photographs of disordered packings of polystyrene particles, showing the structural colors that arise. The radii of the particles increase from left to right. (D) SEM of a disordered packing of 138-nm-radius polystyrene particles. (E) Schematic of the geometry used in our multiple-scattering model. The model is parameterized in terms of experimentally measurable quantities: the volume fraction, complex index of refraction, and radius of the spheres; the complex index of the matrix they are embedded in; the thickness of the film; and the index of refraction of the medium that lies between the viewer and the sample. (F) We calculate the reflectance spectrum by simulating thousands of photon trajectories, a few of which are shown schematically here. (G) The model can predict reflectance spectra that quantitatively agree with experimental measurements, as shown in the plot at Center. Gray area is the uncertainty in the measurement. At Right is a photograph of the measured sample. (H) The model can, therefore, be used to determine the design space—or all of the possible colors—for specific constraints, such as a given type of particle and matrix material. Shown are examples of the colors that can be obtained for fixed material parameters and variable structural parameters. (I) Then, given a target color that is inside the design space, we use optimization to determine the experimental parameters needed to make that color, subject to constraints of our choice, as shown in this schematic. applications, the nanostructure might be too difficult or expen- matrix phase, and we show that it can make predictions that sive to fabricate. are in quantitative agreement with experiment. We then use the Here, we present a way to solve a more common and chal- model to determine the design space, or all of the possible col- lenging design problem: Given the materials and a simple, ors that can be made, given the experimental constraints. We easy-to-make nanostructure—spherical inclusions in a matrix— demonstrate the design of target colors in two ways: In the first, determine what colors can be made and what structural parame- we choose target colors from the design space for specific mate- ters (particle size and volume fraction, for example) are required rial systems. In the second, we target a given point in a perceptual to make a given color. Compared to the problem of determining colorspace and use optimization to determine the experimental the materials and structure required to make a given color, our parameters that produce this color. Overall, the approach that problem is complicated by the potential absence of solutions that we demonstrate (illustrated in Fig.
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